[Forum SIS] Seminario Samantha Leorato - Dip. Scienze Statistiche - 25 maggio 2011

[Paolo Giordani]

25 maggio 2011 - ore 15.00 - Sala 34 (IV piano)

Dipartimento di Scienze Statistiche, Sapienza Università di Roma

Speaker: Samantha Leorato, Università di Roma "Tor Vergata"

Titolo: On a generalization of Jensen inequality for convex functionals

Abstract:
Let $f: \mathcal{X}\mapsto \mathbb{R}$ be a convex mapping and
$\mathcal{X}$ a Hilbert space. The following refinement of Jensen's
inequality is proved: for every convex subsets $A,B\subset \mathcal{X}$
such that $E(X \mid X \in A)=\E(X\mid X \in B)$ and $B\subseteq A$, it
holds $E(f(X)\mid X \in A)\geq E(f(X)\mid X \in B)$. Expectations of
Hilbert-space-valued random elements are defined by means of the Pettis
integrals. Our result generalizes a result of S. Karlin, A. Novikoff,
who derived it for functions with one argument. The inverse implication
is also true if P is an absolutely continuous probability measure. A
convexity criterion based on the Jensen-type inequalities follows.


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Paolo Giordani
Department of Statistical Sciences
Sapienza University of Rome
P.le Aldo Moro, 5 – 00185 Rome, Italy
E-mail: paolo.giordani a uniroma1.it
Tel.: +39 06 49910504, Fax: +39 06 4959241
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